In mathematics, an n-sphere or hypersphere is an ⁠${\displaystyle n}$⁠-dimensional generalization of the ⁠${\displaystyle 1}$⁠-dimensional circle and ⁠${\displaystyle 2}$⁠-dimensional sphere to any non-negative integer ⁠${\displaystyle n}$⁠.

The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general ⁠${\displaystyle n}$⁠-sphere is embedded in an ⁠${\displaystyle n+1}$⁠-dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension ⁠${\displaystyle n\geq 3}$⁠ which are thus embedded in a space of dimension ⁠${\displaystyle n+1\geq 4}$⁠, which means that they cannot be easily visualized. The ⁠${\displaystyle n}$⁠-sphere is the setting for ⁠${\displaystyle n}$⁠-dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in ⁠${\displaystyle (n+1)}$⁠-dimensional Euclidean space, an ⁠${\displaystyle n}$⁠-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an ⁠${\displaystyle (n+1)}$⁠-dimensional ball. In particular:

Given a Cartesian coordinate system, the unit ⁠${\displaystyle n}$⁠-sphere of radius ⁠${\displaystyle 1}$⁠ can be defined as:

Considered intrinsically, when ⁠${\displaystyle n\geq 1}$⁠, the ⁠${\displaystyle n}$⁠-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the ⁠${\displaystyle n}$⁠-sphere are called great circles.

The stereographic projection maps the ⁠${\displaystyle n}$⁠-sphere onto ⁠${\displaystyle n}$⁠-space with a single adjoined point at infinity; under the metric thereby defined, ${\displaystyle \mathbb {R} ^{n}\cup \{\infty \}}$ is a model for the ⁠${\displaystyle n}$⁠-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit ⁠${\displaystyle n}$⁠-sphere is called an ⁠${\displaystyle n}$⁠-sphere. Under inverse stereographic projection, the ⁠${\displaystyle n}$⁠-sphere is the one-point compactification of ⁠${\displaystyle n}$⁠-space. The ⁠${\displaystyle n}$⁠-spheres admit several other topological descriptions: for example, they can be constructed by gluing two ⁠${\displaystyle n}$⁠-dimensional spaces together, by identifying the boundary of an ⁠${\displaystyle n}$⁠-cube with a point, or (inductively) by forming the suspension of an ⁠${\displaystyle (n-1)}$⁠-sphere. When ⁠${\displaystyle n\geq 2}$⁠ it is simply connected; the ⁠${\displaystyle 1}$⁠-sphere (circle) is not simply connected; the ⁠${\displaystyle 0}$⁠-sphere is not even connected, consisting of two discrete points.

For any natural number ⁠${\displaystyle n}$⁠, an ⁠${\displaystyle n}$⁠-sphere of radius ⁠${\displaystyle r}$⁠ is defined as the set of points in ⁠${\displaystyle (n+1)}$⁠-dimensional Euclidean space that are at distance ⁠${\displaystyle r}$⁠ from some fixed point ⁠${\displaystyle \mathbf {c} }$⁠, where ⁠${\displaystyle r}$⁠ may be any positive real number and where ⁠${\displaystyle \mathbf {c} }$⁠ may be any point in ⁠${\displaystyle (n+1)}$⁠-dimensional space. In particular: